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arxiv: 2504.19642 · v2 · submitted 2025-04-28 · 🧮 math.FA

Primal and dual characterizations of sign-symmetric norms

Nguyen Duy Cuong This is my paper

Pith reviewed 2026-05-22 19:11 UTC · model grok-4.3

classification 🧮 math.FA
keywords sign-symmetric normsproduct vector spacesdual normconvex subdifferentialvon Neumann-Jordan constantClarkson resultnormed spaces
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The pith

Sign-symmetric norms on product spaces correspond to convex functions and admit explicit dual and subdifferential formulas.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that sign-symmetric norms on product vector spaces correspond to a class of convex functions. From this link the authors derive closed-form expressions for the dual norm and for the convex subdifferential of any given primal norm. These formulas make the norms convenient for studying geometric and analytic questions on product spaces. As a concrete application the work computes the von Neumann-Jordan constant for such norms and extends a classical identity of Clarkson from Lebesgue spaces to arbitrary normed vector spaces.

Core claim

Sign-symmetric norms on product vector spaces are in one-to-one correspondence with a class of convex functions; this correspondence supplies explicit formulas for the dual norm and the convex subdifferential of the primal norm. The same framework is used to evaluate the von Neumann-Jordan constant of norms on product spaces and thereby extends Clarkson’s result from Lebesgue spaces to general normed vector spaces.

What carries the argument

The sign-symmetry condition on norms defined on product vector spaces, which produces the correspondence with convex functions and the resulting explicit dual and subdifferential expressions.

Load-bearing premise

The norms must be invariant under independent sign flips of the coordinates in the product space.

What would settle it

An explicit sign-symmetric norm whose dual norm, computed directly, fails to match the closed-form expression given in the paper.

read the original abstract

The paper studies primal and dual characterizations of a class of sign-symmetric norms on product vector spaces. Correspondences between these norms and a class of convex functions are established. Explicit formulas for the dual norm and the convex subdifferential of a given primal norm are derived. It is demonstrated that this class of norms is well-suited for studying properties and problems on product spaces. As an application, we study the von Neumann-Jordan constant of norms on product spaces and extend a classical result of Clarkson from Lebesgue spaces to general normed vector spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies sign-symmetric norms on product vector spaces, establishing correspondences with a class of convex functions. It derives explicit formulas for the dual norm and the convex subdifferential of a given primal norm. The class is shown to be well-suited for problems on product spaces. As an application, the von Neumann-Jordan constant is studied on product spaces and a classical result of Clarkson is extended from Lebesgue spaces to general normed vector spaces.

Significance. If the derivations hold, the explicit dual and subdifferential formulas under the sign-symmetry hypothesis provide concrete tools for analyzing norms on product spaces, which is a strength for applications in Banach space geometry. The extension of Clarkson's result to general spaces is a clear positive contribution that broadens the reach of such constants beyond L_p settings. The upfront structural assumption of sign-symmetry avoids hidden circularity and supports focused, verifiable derivations.

minor comments (3)
  1. In the introduction, add a brief concrete example of a sign-symmetric norm on a product space (e.g., R^2 with a weighted l1-type norm) to clarify the property for readers new to the setting.
  2. §3 (dual formula): verify that the explicit dual-norm expression is stated with all variables defined before first use; the current notation for the product-space components is slightly ambiguous on first reading.
  3. The references section contains two minor formatting inconsistencies (missing volume numbers for two journal articles); these should be corrected for consistency with journal style.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation for minor revision. The referee's summary correctly identifies the core contributions: primal-dual characterizations of sign-symmetric norms on product spaces, explicit formulas for the dual norm and subdifferential, and the extension of Clarkson's result on the von Neumann-Jordan constant from Lebesgue spaces to general normed vector spaces. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations self-contained under explicit sign-symmetry hypothesis

full rationale

The paper defines the class of sign-symmetric norms on product spaces as the object of study and establishes direct correspondences with convex functions. From this structural assumption it derives explicit formulas for the dual norm and convex subdifferential. These steps are standard functional-analytic constructions (norm duals, subdifferentials) applied inside the given class rather than reductions of outputs to fitted inputs or self-definitions. The application to the von Neumann-Jordan constant and the extension of Clarkson's result are presented as consequences of the established correspondences, not as independent predictions that collapse back to the inputs by construction. No self-citation load-bearing steps, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation appear in the abstract or description. The derivation chain therefore remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the definition of sign-symmetric norms and standard facts from convex analysis and duality in normed spaces; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Basic properties of norms, dual norms, and convex subdifferentials hold in product vector spaces.
    Invoked to derive explicit formulas and the correspondence with convex functions.

pith-pipeline@v0.9.0 · 5600 in / 1348 out tokens · 36385 ms · 2026-05-22T19:11:17.693066+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A von Neumann-Jordan Constant of Non-Normable Metrics

    math.FA 2026-05 unverdicted novelty 6.0

    Generalized von Neumann-Jordan constant studied for non-normable metrics with validity conditions, examples, counterexamples, and exact formulas for p-metrics on product spaces under a metric-type Clarkson inequality.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [1]

    DOI: 10.1017/CB O9780511662515

    Bonsall, F.F., Duncan, J.: Numerical Ranges II, London Mathematical So ciety Lecture Note Series, Cambridge University Press (1973). DOI: 10.1017/CB O9780511662515

    work page doi:10.1017/cb 1973

  2. [2]

    Acta Math

    Cao, H.X.: V on Neumann-Jordan constants of absolute normalized nor ms on Cn. Acta Math. Sin. 19(3), 507–512 (2003). DOI: 10.1007/s10114-003-0272-4

    work page doi:10.1007/s10114-003-0272-4 2003

  3. [3]

    Clarkson, J.A.: Uniformly convex spaces. Trans. Amer. Math. Soc. 40(3), 396–414 (1936). DOI: 10.1090/S0002-9947-1936-1501880-4

    work page doi:10.1090/s0002-9947-1936-1501880-4 1936

  4. [4]

    Clarkson, J.A.: The von Neumann-Jordan constant for the Lebesgue spaces. Ann. Math. 38, 114–115 (1937). DOI: 10.2307/1968512 20

    work page doi:10.2307/1968512 1937

  5. [5]

    Preprint (2024)

    Cuong, N.D., Kruger, A.Y .: Generalized separation of collections of s ets. Preprint (2024). DOI: 10.48550/arXiv.2412.05336

    work page doi:10.48550/arxiv.2412.05336 2024

  6. [6]

    Dhompongsa, S., Kaewkhao, A., Saejung, S.: Uniform smoothness and U-convexity and ψ-direct sums. J. Nonlinear Convex Anal. 6(2), 327–338 (2005)

    work page 2005

  7. [7]

    A View from V ariational Analysis, 2 edn, Springer, New Y ork (2014)

    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solut ion Mappings. A View from V ariational Analysis, 2 edn, Springer, New Y ork (2014). DOI: 1 0.1007/978-1- 4939-1037-3

    work page 2014

  8. [8]

    Theory and Applications, Springer (2017)

    Ioffe, A.D.: V ariational Analysis of Regular Mappings. Theory and Applications, Springer (2017). DOI: 10.1007/978-3-319-64277-2

    work page doi:10.1007/978-3-319-64277-2 2017

  9. [9]

    Jordan, P ., Neumann, J.V .: On inner products in linear, metric spaces. Ann. Math. 36(3), 719–723 (1935). DOI: 10.2307/1968653

    work page doi:10.2307/1968653 1935

  10. [10]

    Kato, M., Saito, K.S., Tamura, T.: On ψ-direct sums of Banach spaces and convexity. J. Aust. Math. Soc. 75(3), 413–422 (2003). DOI: 10.1017/s1446788700008193

    work page doi:10.1017/s1446788700008193 2003

  11. [11]

    Kato, M., Takahashi, Y .: On the von Neumann-Jordan constant for Banach spaces. Proc. Amer. Math. Soc. 125(4), 1055–1062 (1997). DOI: 10.1090/s0002-9939-97-03740 -4

    work page doi:10.1090/s0002-9939-97-03740 1997

  12. [12]

    Kruger, A.Y ., Mordukhovich, B.S.: Extremal points and the Eu ler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24(8), 684–687 (1980). In Russian. Available from: https://asterius.federation.edu.au/akruger/research/publications.html

    work page 1980

  13. [13]

    Loomis, L., Sternberg, S.: Advanced Calculus, Jones and Bartlett Pu blishers (1990)

    work page 1990

  14. [14]

    Milman, M.: Complex interpolation and geometry of Banach spaces. Ann . Mat. Pura Appl. 136(1), 317–328 (1984). DOI: 10.1007/bf01773388

    work page doi:10.1007/bf01773388 1984

  15. [15]

    Mitani, K.I., Oshiro, S., Saito, K.S.: Smoothness of ψ-direct sum of Banach spaces. Math. Inequal. Appl. 8(1), 147–157 (2005). DOI: 10.7153/mia-08-14

    work page doi:10.7153/mia-08-14 2005

  16. [16]

    Mitani, K.I., Saito, K.S., Suzuki, T.: Smoothness of absolut e norms on Cn. J. Convex Anal. 10(1), 89–107 (2003)

    work page 2003

  17. [17]

    V olu me I: Basic Theory, Springer (2022)

    Mordukhovich, B., Nam, N.: Convex Analysis and Beyond. V olu me I: Basic Theory, Springer (2022). DOI: 10.1007/978-3-030-94785-9

    work page doi:10.1007/978-3-030-94785-9 2022

  18. [18]

    https://doi.org/10.1007/3-540- 45619-8_3

    Mordukhovich, B.S.: V ariational Analysis and Generalized Different iation. I: Basic Theory, vol. 330 Springer, Berlin (2006). DOI: 10.1007/3-540- 31247-1

    work page doi:10.1007/3-540- 2006

  19. [19]

    Persson, L.E., Takahashi, Y ., Kato, M.: Clarkson type inequalit ies and their relations to the concepts of type and cotype. Collect. Math. 51(3), 327–346 (2000)

    work page 2000

  20. [20]

    DOI: 10.1007/978-3-642-02431-3

    Rockafellar, R.T., Wets, R.J.B.: V ariational Analysis, Springer , Berlin (1998). DOI: 10.1007/978-3-642-02431-3

    work page doi:10.1007/978-3-642-02431-3 1998

  21. [21]

    Saito, K.S., Kato, M., Takahashi, Y .: Absolute norms onCn. J. Math. Anal. Appl. 252(2), 879–905 (2000). DOI: 10.1006/jmaa.2000.7139 21

    work page doi:10.1006/jmaa.2000.7139 2000